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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.4.43d
Chapter 10, Problem 10.4.43d

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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Identify the series given: it is a geometric series with the general term \( a_k = \frac{1}{3^k} \) for \( k = 1, 2, 3, \ldots \).
Recall the formula for the sum of an infinite geometric series starting at \( k=1 \): \[ S = \frac{a_1}{1 - r} \] where \( a_1 \) is the first term and \( r \) is the common ratio.
Calculate the first term \( a_1 = \frac{1}{3^1} = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \).
Find the partial sum of the first 10 terms using the formula for the sum of the first \( n \) terms of a geometric series: \[ S_{10} = a_1 \frac{1 - r^{10}}{1 - r} \]. This gives the approximation of the series using 10 terms.
Use the remainder estimate for a geometric series to find the interval where the true sum lies: the remainder \( R_{10} = S - S_{10} = \frac{a_1 r^{10}}{1 - r} \). Since the series is positive and decreasing, the true sum lies between \( S_{10} \) and \( S_{10} + R_{10} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. For |r| < 1, the infinite series converges to a finite sum S = a / (1 - r), where a is the first term. Understanding this helps identify the sum of the given series ∑ 1/3^k.
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Geometric Series

Partial Sums and Approximation

A partial sum is the sum of the first n terms of a series and serves as an approximation to the infinite sum. Using partial sums allows us to estimate the series value by truncating after a finite number of terms, which is essential when calculating the sum up to ten terms.
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Remainder (Error) Estimation in Geometric Series

The remainder after n terms in a geometric series is the difference between the infinite sum and the partial sum. It can be calculated as R_n = S - S_n = a * r^n / (1 - r), providing an interval where the true sum lies. This concept is crucial for bounding the series value after ten terms.
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Related Practice
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

Textbook Question

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.



c.Find the limit of the sequence. What is the physical meaning of this limit?

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.


{1, 2, 4, 8, 16, ......}

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.